How to choose the best iterative strategy for symmetric Toeplitz systems

In the last decade many efficient iterative solvers for n × n symmetric (also Hermitian or generic) Toeplitz systems have been devised [R. H. Chan and G. Strang, SIAM J. Sci. Stat. Comput., 10 (1989), pp. 104-119; T. F. Chan, SIAM J. Sci. Stat. Comput., 9 (1988), pp. 766-771; D. Bini and F. Di Benedetto, Proceedings, 2nd SPAA Conference, 1990, pp. 220-223; D. Bini and P. Favati, SIAM J. Matrix Anal. Appl., 14 (1993), pp. 500-507; T. Huckle, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 767-777; F. Di Benedetto, G. Fiorentino, and S. Serra, Comput. Math. Appl., 26 (1993), pp. 35-45; S. Serra, SIAM J. Matrix Anal. Appl., 17 (1996), pp. 1007-1019; G. Fiorentino and S. Serra, Calcolo, 28 (1991), pp. 283-305; S. Serra, SIAM J. Matrix Anal. Appl., 20 (1998), pp. 31-44]. In all these methods it is assumed that the considered systems have the form An(f)x = b where the symbol f, the generating function, is an L1 function and the entries of An(f) along the kth diagonal coincide with the kth Fourier coefficient of f. The proof of convergence and sometimes even the definition of these solvers are derived by using some symbolic or analytic properties of f, while this function is often unknown in real applications. In this paper, by means of the knowledge of the coefficients of An(f) only, we propose and discuss an algorithm to economically determine the minimal features of f allowing us to choose and define the best iterative strategy. As a consequence of the analysis of the proposed procedure, we find some results on the structure of "quasi algebra" of the set {An(f)}f∈L1. Finally, we exhibit several numerical experiments which confirm the effectiveness of the proposed idea.